Exciton Absorption: Discrete Peaks & Enhanced Continuum

Hello! Welcome back to our course on exciton physics.

In the last two lessons, we developed the tools to understand the optical creation of excitons. We first used Fermi's golden rule to show that the absorption coefficient is proportional to the probability of finding the electron and hole at the same position, $∣ ϕ_{λ} (0) ∣^{2}$ . We then formalized this transition strength into a dimensionless quantity, the oscillator strength, and saw how it scales as $1/ n^{3}$ for the discrete Rydberg series of allowed $s$ -excitons.

So far, we have focused on the sharp absorption peaks corresponding to these discrete, bound exciton states below the bandgap. But what happens to the absorption above the bandgap? Today, we will unite these two pictures. Our goal is to analyze the Elliott-Toyozawa formula for excitonic absorption spectra, including both discrete peaks and the enhanced continuum. This powerful formula provides a complete description of how the electron-hole Coulomb attraction shapes the entire optical absorption spectrum of a semiconductor.

1. From Bound States to the Continuum

Our starting point, as always, is the Wannier equation for the exciton's relative motion. In Module 2, we solved it for the case where the total energy $E$ is less than the bandgap $E_{g}$ , yielding a discrete set of bound states with negative relative energies, $E - E_{g} = - R y^{*} / n^{2}$ .

Now, we must consider the case where the incoming photon has enough energy to create an electron-hole pair with positive relative kinetic energy, i.e., $ℏ ω > E_{g}$ . In this regime, the electron and hole are unbound and can move freely relative to each other, forming a continuum of states.

However, even though they are unbound, they still feel their mutual Coulomb attraction. This attraction influences their wavefunctions. The core task is to find the continuum wavefunctions $ϕ_{E} (r)$ and, specifically, their value at the origin, $∣ ϕ_{E} (0) ∣^{2}$ , since this quantity dictates the absorption strength.

The derivation involves solving the Schrödinger equation for a scattering problem in a Coulomb potential, which is a standard problem in quantum mechanics. While we won't reproduce the full derivation, we will use a resource that outlines the key steps and the crucial result.

10. Excitons in Bulk and Two-dimensional Semiconductors

The key to unifying the discrete and continuous parts of the spectrum is to solve the Wannier equation for both bound and unbound states. This resource provides a concise derivation of the complete absorption spectrum for a 3D semiconductor.

Please read the first part of the document, up to and including the derivation of the '3D Elliott formula' (the first ~2 pages). Focus on these key steps: The setup of the Wannier equation in normalized exciton units. The expression for the bound state wavefunctions at the origin. The treatment of continuum states using box normalization and the resulting expression for the wavefunction at the origin. The important result is the final expression for the wave function at the origin for continuum states. The final summation over discrete and continuum states to arrive at the full expression for the imaginary part of the susceptibility, Im(χ).

2. The Elliott-Toyozawa Formula

As you saw in the reading, the total absorption is found by summing the contributions from all possible final states—both the discrete bound states and the continuous unbound states. This leads to the Elliott-Toyozawa formula.

Let's break down the result derived in the resource. The imaginary part of the dielectric function (or susceptibility), which is proportional to the absorption coefficient $α (ω)$ , can be written as the sum of two parts:

Im [ϵ (ω)] \propto α (ω) \propto n = 1 \sum \infty \frac{1}{( a _{B} ) ^{3} n ^{3}} δ (ℏ ω - (E_{g} - \frac{R y ^{*}}{n ^{2}})) + \frac{θ ( ℏ ω - E _{g} )}{1 - e ^{- 2 π R y^{*} / (ℏ ω - E_{g})}}

Let's analyze this fundamental result piece by piece.

a) The Discrete Spectrum: Below the Bandgap ( $ℏ ω < E_{g}$ )

The first term describes the absorption below the bandgap:

\propto n = 1 \sum \infty \frac{1}{n ^{3}} δ (ℏ ω - E_{n})

This is a series of delta functions, representing infinitesimally sharp absorption lines at the exciton energies $E_{n} = E_{g} - R y^{*} / n^{2}$ .
The weight of each peak, or its oscillator strength, is proportional to $1/ n^{3}$ . This confirms the result from our last lesson, which we derived from the hydrogenic wavefunction $∣ ϕ_{n s} (0) ∣^{2} = 1/ (π (n a_{B})^{3})$ . The $1 s$ exciton state has the highest probability of electron-hole overlap and thus the strongest absorption.

b) The Continuous Spectrum: Above the Bandgap ( $ℏ ω > E_{g}$ )

The second term describes the absorption for energies above the bandgap:

\propto \frac{1}{1 - e ^{- 2 π η}} where η = \frac{R y ^{*}}{ℏ ω - E _{g}}

The Heaviside step function $θ (ℏ ω - E_{g})$ simply ensures this term is zero below the bandgap.
The factor $1/ (1 - e^{- 2 π η})$ is known as the Sommerfeld enhancement factor. It quantifies how the Coulomb attraction modifies the absorption continuum compared to that of free, non-interacting electron-hole pairs.

Let's examine the behavior of this continuum absorption.

Sommerfeld Enhancement in Excitonic Absorption — Optical absorption spectrum illustrating the key features of the Elliott-Toyozawa formula. Below the bandgap energy $E_g$, discrete absorption peaks correspond to the formation of bound exciton states (1s, 2s, etc.). Above $E_g$, the absorption is a continuum of states, but its magnitude is significantly enhanced by the Coulomb interaction between the electron and hole (the "Sommerfeld enhancement"). The dashed line shows the $\sqrt{\hbar\omega - E_g}$ dependence expected for free electron-hole pairs, which starts at zero at the band edge. In contrast, the excitonic continuum absorption starts at a finite value.

At the band edge ( $ℏ ω \to E_{g}^{+}$ ):
The parameter $η \to \infty$ . In this limit, $e^{- 2 π η} \to 0$ , so the Sommerfeld factor approaches 1. This is a profound result. For free electron-hole pairs, the joint density of states goes as $ℏ ω - E_{g}$ , meaning the absorption starts at zero and gradually increases. The Coulomb attraction, however, "piles up" the states at the band edge, causing the absorption to jump to a finite, constant value right at $E_{g}$ .

Far above the band edge ( $ℏ ω ≫ E_{g}$ ):
The parameter $η \to 0$ . We can expand the exponential for small arguments: $e^{- 2 π η} \approx 1 - 2 π η$ . The Sommerfeld factor becomes $1/ (1 - (1 - 2 π η)) = 1/ (2 π η)$ . The full continuum absorption, which includes the density of states, will then behave like the free-particle case. Physically, at very high energies, the electron and hole are moving so fast that their mutual attraction becomes negligible, and they behave like free particles.

Test your understanding!

Consider the limit where the exciton binding energy $R y^{*}$ goes to zero (e.g., in a material with extremely high dielectric screening). What does the Elliott-Toyozawa formula predict for the absorption spectrum in this limit?

Show answer

If $R y^{*} \to 0$ , two things happen:

The discrete part vanishes because all binding energies go to zero, so no bound states can form below $E_{g}$ .
In the continuum part, the parameter $η = R y^{*} / (ℏ ω - E_{g}) \to 0$ for all $ℏ ω > E_{g}$ . As we saw, in this limit the Sommerfeld enhancement disappears. The absorption would then follow the free-particle behavior, which is governed by the joint density of states: $α (ω) \propto ℏ ω - E_{g}$ . The formula correctly recovers the independent particle model when Coulomb effects are negligible.

3. Connection to Experiment

The Elliott-Toyozawa formula provides a remarkably accurate description of the absorption spectra of many direct-gap semiconductors. Of course, in real materials, the delta-function peaks are broadened into finite-width profiles (e.g., Lorentzians or Gaussians) due to finite exciton lifetimes, interactions with phonons, and crystal imperfections.

Your research focus on Cu₂O provides a spectacular experimental confirmation of the discrete part of the Elliott formula. While the transitions are parity-forbidden and it's the P-excitons that are optically active (as we discussed last lesson), the spectrum still consists of a beautiful Rydberg series of lines converging on the bandgap.

The following paper shows this with exceptional clarity.

Giant Rydberg Excitons in Cuprous Oxide

Let's connect this theoretical formula to a real-world spectrum. You are familiar with Cu₂O, which shows a spectacular example of the discrete exciton series. This paper on Rydberg excitons provides one of the cleanest experimental demonstrations.

Please briefly review this paper, focusing on the following aspects: Figure 1a: Look at the absorption spectrum showing the P-exciton series up to n=25. This is the experimental manifestation of the discrete sum in the Elliott formula. Section starting 'The measured absorption lines...': Notice how the authors fit the lines to determine the resonance energies E_n. They confirm the E_n = E_g - Ry^*/(n-\delta_P)^2 dependence. This is a slight modification of our ideal hydrogenic formula to account for real-crystal effects via the 'quantum defect' \delta_P, a concept you may know from atomic physics. Supplementary Information ('Corrections of nP resonances...'): Skim the short paragraph around Eq. S1. It shows the fitting formula used to account for line asymmetry due to phonon interactions (Fano resonance), a real-world complication to our ideal theory.

The spectra in this paper beautifully illustrate the discrete series of peaks that form the first part of the Elliott-Toyozawa formula. The ability to resolve states up to $n = 25$ is a testament to the high quality of the Cu₂O crystal and the robustness of the Wannier-Mott exciton model.

Conclusion

In this lesson, we have unified our understanding of excitonic optical absorption. We've moved beyond considering only the discrete peaks to a complete picture that includes the continuous absorption above the bandgap.

Key Takeaways:

The Elliott-Toyozawa formula describes the entire absorption spectrum of a direct-gap semiconductor, incorporating both the discrete Rydberg series of bound excitons and the enhanced continuum of unbound electron-hole pairs.
The Coulomb interaction is responsible for both phenomena: it binds the electron and hole into discrete states below $E_{g}$ and enhances the absorption probability for states in the continuum just above $E_{g}$ .
This Sommerfeld enhancement causes the absorption to begin with a finite step at the band edge, in stark contrast to the $ℏ ω - E_{g}$ dependence of free particles.
At high energies, the Coulomb attraction becomes negligible, and the excitonic absorption spectrum smoothly transitions to the free-particle behavior.

Preview of the Next Lesson

We have now established a complete model for the absorption line shape. The next logical step is to delve deeper into which transitions are allowed in the first place. In the next lesson, we will derive the basic optical selection rules for exciton transitions based on parity and angular momentum conservation. This will formalize our discussion of S-excitons in GaAs versus P-excitons in Cu₂O and provide the group-theoretical foundation for interpreting complex spectra.

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